ECE331 HOMEWORK 8

(0)[reading corresponding with the lectures] 
Read Yates chapters 8.1, 8.2, 11.2, 10.1-10.5
Also reading the lecture notes can help with this homework.

(1) Yates 6.5.1

(2) Yates 6.5.4

(3) Yates 11.2.3

(4) Yates 6.7.8

(5) Let M(t) and N(t) be independent Poisson processes with rates
a and b respectively.
Prove that X(t)=M(t)+N(t) is a Poisson process of rate a+b.

(6) Yates 10.1.1

(7) State and prove the Markov inequality and the Chebyshev 
inequality. (see section 8.2 of yates)

(8) Let X(t) be the random telegraph process so that 
X(0) is +1 or -1 with equal probability and 
X(t) changes sign at the events of a Poisson process of rate a.
Give a full derivation of the mean and autocorrelation of X(t).
(note: this will be covered in Tuesdays lecture)

(9) Let X(t)=cos(t+U) where U is uniform on [0,2pi).
Thus X(t) is a sinusoid with random phase.
Compute the mean and autocorrelation of X(t).

(10) State the central limit theorem.
Using the central limit theorem, explain the construction of
the Wiener process as the limit of a symmetric random walk.